Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Diffusion equations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 361 records · Page 20
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

A Preliminary Overview of Metal Fuel Models in BISON

BISON is a modern nuclear fuel performance code that has been under development at Idaho National Laboratory (INL) since 2009. It is a finite element-based code capable of solving the energy, mechanics, and species diffusion equations in a fully-coupled sense. BISON also utilizes thermal and mechanical contact algorithms and variety of nonlinear material models. BISON is applicable to steady and transient fuel behavior and supports the use of one-, two-, and three-dimensional meshes. Although thus far used primarily to simulate light water reactor fuel, BISON is designed to be a multi-fuels code. As such, BISON has been used to analyze TRISO-coated particle fuel and metal fuel in rod and plate form, design and interpret fuel irradiation experiments and investigate novel fuel concepts. This publication provides a preliminary list of metal fuel-related material models recently implemented in BISON. Some preliminary results from a simulation of the EBR-II X447 experiment are shown, which begins to demonstrate how all these models work together. A follow-up publication will be presented with more detail at the ANS meeting in New Orleans during the embedded topical meeting titled 'Nuclear Fuels and Structural Materials for the Next Generation Nuclear Reactors'. (authors)

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2020

An Improved Model For Determining Salinity Recharge Time for Deep Borehole Disposal - 20405

In a previous publication (K. P. Travis, D. Burley and F. G. F. Gibb, WM2017 Conference, Phoenix Arizona paper no. 17480) we introduced a numerical model to estimate the time it would take for a pressure perturbation to subside following its creation from the construction of a deep geological borehole in water-saturated rock. The model was based on the notion of a point source of momentum - a solution of the time-dependent pressure-diffusion equation. Using the model, we estimated that it would take on the order of 10 k years for physical equilibrium to become re-established following the sinking of a 5 km borehole in granite. The significance of such a model is that it places an upper bound on the lifetime required of a borehole sealing system - a necessary input to a borehole post-closure safety case assessment. No engineered sealing system has ever been devised which is capable of retaining its sealing properties for 10 half-lives of long-lived radioisotopes in spent fuel or high-level waste. One of the key advantages of disposing of nuclear waste via Deep Borehole Disposal (DBD) is the natural sealing provided by density stratified groundwater. Upon sinking of a borehole, and subsequent filling with fresh water, brine or drilling mud, this natural barrier may be temporarily damaged. Over time, fresh brine from the far-field will flow towards or away from the hole (driven by a pressure gradient) and will eventually re-establish the original salinity gradient. It follows that the engineered seals need last only as long as the time required for this salinity gradient to reset itself. One of the limitations of our previous model was the use of a static boundary condition on the borehole wall. The model used a boundary pressure which varied quadratically with depth (arising from differences between the pressure of a column of fresh water in the borehole and that of a column of brine in the host rock). However as brine replaces fresh water in the borehole (driven by a pressure gradient), the boundary function must change with time. We now introduce an improved model which takes this time dependent boundary condition into account. The model also takes into account the time taken for a mixture of brine and fresh water to re-establish chemical equilibrium through the process of diffusion. The paper contains the mathematical details of our new iterative model as well as results showing the time taken to reach steady state, and concentration profiles for the components of the brine-filled borehole as a function of time together with a discussion on the implications of the results for developing a post-closure safety assessment for DBD. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2014

Fast penetration of megagauss fields into metallic conductors

Megagauss magnetic-field penetration into a conducting material is studied via a simplified but representative model, wherein the magnetic-diffusion equation is coupled with a thermal-energy balance. The specific scenario considered is that of a prescribed magnetic field rising (in proportion to an arbitrary power r of time) at the surface of a conducting half-space whose electric conductivity is assumed proportional to an arbitrary inverse power γ of temperature. We employ a systematic asymptotic scheme in which the case of a strong surface field corresponds to a singular asymptotic limit. In this limit, the highly magnetized and hot “skin” terminates at a distinct propagating wave-front. Employing the method of matched asymptotic expansions, we find self-similar solutions of the magnetized region which match a narrow boundary-layer region about the advancing wave front. The rapidly decaying magnetic-field profile in the latter region is also self similar; when scaled by the instantaneous propagation speed, its shape is time-invariant, depending only on the parameter γ. The analysis furnishes a simple asymptotic formula for the skin-depth (i.e., the wave-front position), which substantially generalizes existing approximations. It scales with the power γr + 1∕2 of time and the power γ of field strength, and is much larger than the field-independent skin depth predicted by an athermal model. The formula further involves a dimensionless O(1) pre-factor which depends on r and γ. It is determined by solving a nonlinear eigenvalue problem governing the magnetized region. Another main result of the analysis, apparently unprecedented, is an asymptotic formula for the magnitude of the current-density peak characterizing the wave-front region. Complementary to these systematic results, we provide a closed-form but ad hoc generalization of the theory approximately applicable to arbitrary monotonically rising surface fields. Our results are in excellent agreement with numerical simulations of the model, and compare favourably with detailed magnetohydrodynamic simulations reported in the literature.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

NMR_DIFFSIM v. 1.0

The software generates and analyzes nuclear magnetic resonance spin diffusion simulations for the purpose of measuring domain sizes or spin diffusion constants in a heterogenous sample. Its primary capabilities are: 1. It solves the spin magnetization diffusion equation to simulate the results of a spin diffusion experiment using any structural input geometry, including structure models generated by molecular dynamics or other computational or experimental methods. 2. It compares the results of simulations on an input structure to experimental data, providing a data fitting algorithm to extract physical information from the experimental data via the simulation. 3. It can provide statistical measures to compare correlations between structural models, to determine domain clustering and connectedness, and to provide goodness-of-fit measures including chord length distributions, chi-squared values, Pearson coefficients and pair correlation functions. 4. It provides output reports, data files, and movies to visualize simulation results.

Sorte, Eric↗
NASA NTRS1975

A cubic spline approximation for problems in fluid mechanics

A cubic spline approximation is presented which is suited for many fluid-mechanics problems. This procedure provides a high degree of accuracy, even with a nonuniform mesh, and leads to an accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several implicit and explicit integration schemes are presented. For two-dimensional flows, a spline-alternating-direction-implicit method is evaluated. The spline procedure is assessed, and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1975

Viscous flow solutions with a cubic spline approximation

A cubic spline approximation is used for the solution of several problems in fluid mechanics. This procedure provides a high degree of accuracy even with a nonuniform mesh, and leads to a more accurate treatment of derivative boundary conditions. The truncation errors and stability limitations of several typical integration schemes are presented. For two-dimensional flows a spline-alternating-direction-implicit (SADI) method is evaluated. The spline procedure is assessed and results are presented for the one-dimensional nonlinear Burgers' equation, as well as the two-dimensional diffusion equation and the vorticity-stream function system describing the viscous flow in a driven cavity. Comparisons are made with analytic solutions for the first two problems and with finite-difference calculations for the cavity flow.

Rubin, S. G.↗
NASA NTRS1976

The absorption of trapped particles by the inner satellites of Jupiter and the radial diffusion coefficient of particle transport

The process of trapped particle absorption by the inner Jovian satellites is considered in detail taking into account both the particle and satellite motions in a magnetic dipole field which is displaced from the center of the planet and tilted with respect to the planetary rotation axis. An expression is derived for computing the sweeping time at a given satellite, defined as the time required for the satellite to sweep up a given fraction of the trapped particles within its sweeping region. By making use of the sweeping time and the radial diffusion equation of particle transport approximate expressions for the diffusion coefficient are derived. Measurements obtained by Pioneer 10 are then used to obtain estimates of the diffusion coefficient at the orbits of Io and Europa. We find that the diffusion coefficient is a function of energy and magnetic latitude for electrons in the energy range 0.7-14 MeV.

Mogro-Campero, A.↗
NASA NTRS1986

Monte Carlo study of a model of diffusion-controlled reactions

The Monte Carlo method is used to perform averages over sink configurations in the present study of diffusion-controlled reactions occurring between solute particles and immobile spherical sinks. In order to determine the average steady state solute concentration profile in a locally perturbed solution for sink volume fractions phi of less than 0.3, the diffusion equation in the monopolar plus dipolar approximation of diffusive couplings between the sinks is solved numerically. The Monte Carlo method is shown to be the most accurate and efficient in the phi = 0.001-0.1 region, where a system of only 25 sinks suffices and the monopolar approximation alone is sufficiently accurate.

Beenakker, C. W. J.↗
NASA NTRS1996

NASA flow fields analysis

The objectives of the present research are to improve design capabilities for low thrust rocket engines through understanding the detailed mixing and combustion processes in a representative combustor. Of particular interest is a small gaseous hydrogen-oxygen thruster which is considered as a coordinated part of an on-going experimental program at NASA LERC. Detailed computational modeling involves the solution of both the two- and three-dimensional Navier Stokes equations, coupled with chemical reactions and the species diffusion equations. Computations of interest include both steady state and time-accurate flowfields and are obtained by means of LU approximate factorization in time and flux split upwinding differencing in space. The emphasis in the research is focused on using numerical analysis to understand detailed combustor flowfields, including the shear layer dynamics created between fuel film cooling and the core gas in the vicinity on the nearby combustor wall; the integrity and effectiveness of the coolant film; and three-dimensional fuel and oxidizer jet injection/mixing/combustion characteristics in the primary combustor along with their joint impacts on global engine performance.

Merkle, Charles L.↗
NASA NTRS1987

Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations

The current status of streamline-upwind/Petrov-Galerkin (SUPG) methods for the analysis of flow problems is surveyed in an analytical review. Problem areas addressed include classical Galerkin, upwind, artificial-diffusion, SUPG, discontinuous Galerkin, space-time FEM, and discontinuity-capturing approaches to the scalar advection-diffusion equation; incompressible flows; advective-diffusive systems; and the compressible Euler and Navier-Stokes equations. Graphs and diagrams are provided, and the good stability properties of state-of-the-art SUPG methods are pointed out.

Hughes, Thomas J. R.↗
DOE OSTI2014

A reconsideration of the noise equivalent power and the data analysis procedure for the infrared imaging video bolometers

The infrared imaging video bolometer (IRVB) used for measurement of the two-dimensional (2D) radiation profiles from the Large Helical Device has been significantly upgraded recently to improve its signal to noise ratio, sensitivity, and calibration, which ultimately provides quantitative measurements of the radiation from the plasma. The reliability of the quantified data needs to be established by various checks. The noise estimates also need to be revised and more realistic values need to be established. It is shown that the 2D heat diffusion equation can be used for estimating the power falling on the IRVB foil, even with a significant amount of spatial variation in the thermal diffusivity across the area of the platinum foil found experimentally during foil calibration. The equation for the noise equivalent power density (NEPD) is re-derived to include the errors in the measurement of the thermophysical and the optical properties of the IRVB foil. The theoretical value estimated using this newly derived equation matches closely, within 5.5%, with the mean experimental value. The change in the contribution of each error term of the NEPD equation with rising foil temperature is also studied and the blackbody term is found to dominate the other terms at elevated operating temperatures. The IRVB foil is also sensitive to the charge exchange (CX) neutrals escaping from the plasma. The CX neutral contribution is estimated to be marginally higher than the noise equivalent power (NEP) of the IRVB. It is also established that the radiation measured by the IRVB originates from the impurity line radiation from the plasma and not from the heated divertor tiles. The change in the power density due to noise reduction measures such as data smoothing and averaging is found to be comparable to the IRVB NEPD. The precautions that need to be considered during background subtraction are also discussed with experimental illustrations. Finally, the analysis algorithm with all the improvements is validated and found to reproduce the input power well within 10% accuracy. This article answers many fundamental questions relevant to the IRVB and illustrates the care to be exercised while processing the IRVB data.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRS1985

Pitch Angle Distributions of Solar Energetic Particles and the Local Scattering Properties of the Interplanetary Magnetic Field

An approximate solution of the Fokker-Planck equation containing pitch angle scattering and adiabatic focusing is discussed. For modest focusing effects the omnidirectional density obeys an ordinary diffusion equation with a modified diffusion coefficient. The anisotropic part of the distribution function is properly normalized and split into an even and an odd part. The even part is determined by the ratio between the scattering mean free path and the focusing length and by the degree of polarization of the magnetic field fluctuations. The odd part is determined by the deviation of the pitch angle scattering from isotropic scattering. The method supplies a powerful tool to obtain the local characteristics of pitch angle scattering. It is insensitive to long lasting solar injections and to moderate radial variations of the mean free path. The method is applied to solar particle events observed on Helio-1 and -2.

Beeck, J.↗
DOE OSTI2023

Evaluation of dual-weighted residual and machine learning error estimation for projection-based reduced-order models of steady partial differential equations

Projection-based reduced-order models (pROMs) show great promise as a means to accelerate many-query applications such as forward error propagation, solving inverse problems, and design optimization. In order to deploy pROMs in the context of high-consequence decision making, accurate error estimates are required to determine the region(s) of applicability in the parameter space. The following paper considers the dual-weighted residual (DWR) error estimate for pROMs and compares it to another promising pROM error estimate, machine learned error models (MLEM). Here, we show how DWR can be applied to ROMs and then evaluate DWR on two partial differential equations (PDEs): a two-dimensional linear convection–reaction–diffusion equation, and a three-dimensional static hyper-elastic beam. It is shown that DWR is able to estimate errors for pROMs extrapolating outside of their training set while MLEM is best suited for pROMs used to interpolate within the pROM training set.

42 ENGINEERING↗
DOE OSTI2017

Physics-based control-oriented modeling of the current density profile evolution in NSTX-Upgrade

Active control of the toroidal current density profile is among those plasma control milestones that the National Spherical Torus eXperiment-Upgrade (NSTX-U) program must achieve to realize its next-step operational goals. Motivated by the coupled, nonlinear, multivariable, distributed-parameter plasma dynamics, the first step towards control design is the development of a physics-based, control-oriented model for the current profile evolution in response to non-inductive current drives and heating systems. The evolution of the toroidal current density profile is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The proposed control-oriented model predicts the spatial-temporal evolution of the current density profile by combining the nonlinear MDE with physics-based correlations obtained at NSTX-U for the electron density, electron temperature, and non-inductive current drives (neutral beams). The resulting first-principles-driven, control-oriented model is tailored for NSTX-U based on predictions by the time-dependent transport code TRANSP. Furthermore, main objectives and possible challenges associated with the use of the developed model for the design of both feedforward and feedback controllers are also discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2019

Robust control of the current profile and plasma energy in EAST

Integrated control of the toroidal current density profile, or alternatively the q-profile, and plasma stored energy is essential to achieve advanced plasma scenarios characterized by high plasma confinement, magnetohydrodynamics stability, and noninductively driven plasma current. The q-profile evolution is closely related to the evolution of the poloidal magnetic flux profile, whose dynamics is modeled by a nonlinear partial differential equation (PDE) referred to as the magnetic-flux diffusion equation (MDE). The MDE prediction depends heavily on the chosen models for the electron temperature, plasma resistivity, and non-inductive current drives. To aid control synthesis, control-oriented models for these plasma quantities are necessary to make the problem tractable. However, a relatively large deviation between the predictions by these control-oriented models and experimental data is not uncommon. For this reason, the electron temperature, plasma resistivity, and non-inductive current drives are modeled for control synthesis in this work as the product of an “uncertain” reference profile and a nonlinear function of the different auxiliary heating and current-drive (H&CD) source powers and the total plasma current. The uncertainties are quantified in such a way that the family of models arising from the modeling process is able to capture the q-profile and plasma stored energy dynamics from a typical EAST shot. A control-oriented nonlinear PDE model is developed by combining the MDE with the “uncertain” models for the electron temperature, plasma resistivity, and non-inductive current drives. This model is then rewritten into a control framework to design a controller that is robust against the modeled uncertainties. The resulting controller utilizes EAST's H&CD powers and total plasma current to regulate the q profile and plasma stored energy even when mismatches between modeled and actual dynamics are present. The effectiveness of the controller is demonstrated through nonlinear simulations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Multigrid Reduction in Time for Nonlinear Parabolic Problems: A Case Study

The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the p-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and p=2 corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficulty encountered is that the nonlinear time-step solver becomes progressively more expensive on coarser time levels as the time-step size increases. To overcome such difficulties, multigrid research has historically targeted an accumulated body of experience regarding how to choose an appropriate solver for a specific problem type. To that end, this paper develops a library of MGRIT optimizations and modifications, most important an alternate initial guess for the nonlinear time-step solver and delayed spatial coarsening, that will allow many nonlinear parabolic problems to be solved with parallel scaling behavior comparable to the corresponding linear problem.

parareal↗
DOE OSTI2021

AIR Algebraic Multigrid for a Space-Time Hybridizable Discontinuous Galerkin Discretization of Advection(-Diffusion)

This article investigates the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable discontinuous Galerkin discretization of advection-dominated flows. The motivation for this study is that the time-dependent advection-diffusion equation can be seen as a “steady” advection-diffusion problem in $(d+1)$-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. Numerical examples demonstrate the effectiveness of AIR as a preconditioner for advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable, space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems.

97 MATHEMATICS AND COMPUTING↗